Question

The Prices of different articles and demand for them are shown in the following frequency distribution table. Find the median of the prices.

Price (Rupees)	20 less than	20-40	40-60	60-80	80-100
No. of articles	140	100	80	60	20

Answer 1

Hint: First, find the cumulative frequency of the articles. After that find the median class. From there we will get the values of the lower limit, frequency, class interval, and preceding cumulative frequency of median class. Then, substitute the values in the median formula $Median=l+\dfrac{\dfrac{N}{2}-c.f.}{f}\times h$ to get the median of the prices.

Complete step by step answer:
Given,

Price (Rupees)	20 less than	20 – 40	40 – 60	60 – 80	80 – 100
No. of articles	140	100	80	60	20

Now, calculate the cumulative frequency of the articles,

Class(Price in Rupees)	Frequency(Number of items)	Cumulative Frequency(Less than the upper limit)
0 – 20	140	140
20 – 40	100	240
40 – 60	80	320
60 – 80	60	380
80 – 100	20	400
	N = 400

The half value of total frequency is,
$\Rightarrow \dfrac{N}{2}=\dfrac{400}{2}=200$
The half value of frequency lies in class 20 – 40. So,
Median Class $= 20 – 40$
Here, we have,
Lower Median Class, $l = 20$
Cumulative frequency, $c.f. = 140$
Frequency, $f = 100$
Width of the interval, $h = 20$
The formula of the median is,
$Median=l+\dfrac{\dfrac{N}{2}-c.f.}{f}\times h$
Substitute the values in the above formula,
$\Rightarrow Median=20+\dfrac{200-140}{100}\times 20$
Subtract the values in the numerator and cancel out the common factors in numerator and denominator,
$\Rightarrow Median=20+\dfrac{60}{5}$
Cancel out the common factor,
$\Rightarrow Median=20+12$
Add the terms on the right side to get the median,
$\Rightarrow Median=32$

Hence, the median of the prices is 32.

Note:
The Mean, Median, and Mode are the three measures of central tendency.
Mean is the arithmetic average of a data set. This is found by adding the numbers in a data set and dividing by the number of observations in the data set.
Median is the middle number in a data set when the numbers are listed in either ascending or descending order.
The mode is the value that occurs the most often in a data set and the range is the difference between the highest and lowest values in a data set.